Instabilities of an annulus flow between rotating cylinders in a helical magnetic field

The stabilities of an annulus flow between the rotating inner and outer cylinders with an external helical magnetic field are studied by using the quasistatic approximation. It is shown numerically that for the spiral base flow with a zero axial pressure gradient, the helical magnetic field yields a helical traveling wave at a critical Reynolds number. This wave mode is revealed to be the most unstable mode by linear stability analysis. At higher Reynolds numbers, the first wave mode is superposed by a second antisymmetric helical wave mode, which travels with a higher phase velocity than the first mode. When the Reynolds number is increased further, the flow becomes turbulent, but the key features of the flow structure are still dominated by the first and the second wave modes. Furthermore, when a finite axial pressure gradient is applied to guarantee a zero axial flow rate, the annulus flow is found to be more unstable than the case with zero axial pressure gradient.

Figure 1

The azimuthal velocity $U$, the axial velocity $V$, and the electric potential $\mathrm{\Phi }$ for the base flows of (a) zero axial pressure gradient (BF1) and (b) zero axial flow rate (BF2). Numerical results obtained at $\text{Re}=500$ (${\text{Re}}_z=86$) and 200 are shown by gray symbols in (a) and (b), respectively, and the black points indicate the maximum and the minimum values on the curves.

Figure 2

(a) The time series of the disturbance kinetic energy at $\text{Re}=650$ (${\text{Re}}_z=111.8$). (b) The mean disturbance kinetic energy as a function of the Reynolds numbers. The solid squares are the solutions calculated with initial random disturbances, while the open squares represent the states obtained by increasing or decreasing Re from saturated solutions.

Figure 3

Flow structure of the first helical wave mode. Instantaneous isosurfaces of the radial velocity $u_r=-0.0113$ between $z=-\pi$ and $\pi$ at (a) $\text{Re}=580$ (${\text{Re}}_z=99.76$) and (b) $\text{Re}=650$ (${\text{Re}}_z=111.8$) are shown with the isocontours of $u_r$ in the cross sections at $z=0$, respectively. The arrows in (a) indicate the directions of the external magnetic fields. The instantaneous isocontours of the azimuthal velocity (c), the azimuthal vorticity (e), the disturbance electric potential (f), as well as the vector plot of the in-plane velocity ($u_r,u_z)$ on a coarse mesh (d), at $\text{Re}=580$ are shown in a part of a meridional plane. Note that in (d) $u_z$ is amplified three times in order to show the flow field clearly.

Figure 4

(a) The instantaneous isosurfaces of $u_r=\pm 0.02$ (light gray-dark gray/yellow-blue in color version) in the part between $z=\pm \pi$ at $\text{Re}=850$ (${\text{Re}}_z=146.2$) accompanied by the isocontours of $u_r$ at $z=0$. The isosurfaces of $u_r=\pm 0.01$ (light gray-dark gray/yellow-blue in color version) at $\text{Re}=600$ are shown in (b) as a reference. The instantaneous isocontours of the azimuthal velocity (c), the azimuthal vorticity (e), the disturbance electric potential (f), as well as the vector plot of the in-plane velocity (d), at $\text{Re}=850$ are shown in a part of a meridional plane.

Figure 5

The azimuthal wave number $k_{\theta }$ (a), the axial wave number $k_z$ (b), the angular frequency $\omega$ (c), and the phase velocity $v$ (d) of the first helical wave mode (solid symbols) and the second helical wave mode (open symbols) at various Reynolds numbers.

Figure 6

Instantaneous isosurfaces of the radial velocity $u_r=0.0195$ (light gray/green in color version) and $u_r=-0.0431$ (dark gray/blue in color version) at $\text{Re}=2000$ (${\text{Re}}_z=344$). The time difference between (a) and (b) is 1.03, and the arrow indicates the rotation direction of the cylinders. (c) The time-averaged azimuthal spectra of the kinetic energy obtained at different Reynolds numbers.

Figure 7

The neutral curves for the case with zero axial pressure gradient in (a) the $k_z\text{−}\text{Re}$ plane and (b) the $\omega \text{−}\text{Re}$ plane, where the circles indicate the critical state revealed in numerical simulations. The black points represent the critical state of the left-handed mode. Contours of the radial velocity $w^{\prime }$ and the disturbance electric potential ${\varphi }^{\prime }$ for the right-handed mode at the critical neutral state $(\text{Re},k_{\theta },k_z,\omega )=(500.23,3,-3.35,0.394)$ are shown in (c) and (e), respectively, and those for the left-handed mode at the critical state of $(804.25,2,6.41,2.43)$ are shown in (d) and (f), respectively. The disturbance parameters are normalized such that the volume-averaged disturbance kinetic energy $E/\left(2V\right)$ is unit.

Figure 8

The neutral curves for the case with zero axial flow rate in (a) the $k_z\text{−}\text{Re}$ plane and (b) the $\omega \text{−}\text{Re}$ plane. Contours of the radial velocity $w^{\prime }$ and the disturbance electric potential ${\varphi }^{\prime }$ of the right-handed, the left-handed, and the axisymmetric modes at the critical states labeled by circles in (a) and (b) $(\text{Re},k_{\theta },k_z,\omega )=(202.47,1,-4.96,-0.64),(334.36,1,9.05,2.37)$, and $(260.56,0,\pm 6.29,\pm 1.31)$ are shown in (c) and (f), (d) and (g), and (e) and (h), respectively. The disturbance parameters are normalized such that the volume-averaged disturbance kinetic energy $E/\left(2V\right)$ is unit.